In The Elder Scrolls V: Skyrim for the PC, PS3, and Xbox 360, Breezehome is one of the first houses you’ll have the opportunity to buy, and it is also the cheapest at a mere 5,000 gold.
To buy Breezehome, you just need to do the primary Whiterun quests for the Jarl there. After you complete the Bleak Falls Barrow quest, Jarl Bulgruuf the Greater will announce that he has made you a Thane, and has granted you the opportunity to purchase a home in Whiterun. Speak to his adviser Proventus Avenicci at this point to purchase Breezehome.
Proventus is also the one you’ll be dealing with to upgrade your home as well.